37s Royal Society/. 



nearly 3"j but if situated at the same angular distance from tl)e 

 ecliptic as Sirius is, it would have an annual parallax in latitude of 

 1-8". 



Dec. 18. — A paper was read, intitled, "An attempt to rectify the 

 inaccuracy of two logarithmic formulae." By J. T. Graves, Esq. ; 

 communicated by J. F. W. Herschel, Esq. V.P.R.S. 



The discovery made by Poisson and Poinsot, during their recent 

 researches on angular sections, of errors in trigonometrical formul;e 

 usually deemed comjilete, drew the attention of the author to an ana- 

 logous incorrectness in logarithmic series. He accordingly proposes, 

 in the present paper, to exhibit, in an amended form, two fundamental 

 developments; the principles employed in the establishment of which 

 admit of application in expanding, by different methods, various simi- 

 lar functions, and tend also to elucidate other parts of the exponential 

 theory. He then enters into an analytical investigation of the equa- 

 tion «''=?/,• and exhibits correct developments ; first, of )/, in terms 

 of a and x ; and secondly, of a-, in terms of a and y ; the correspond- 

 ing developments hitherto given being incomplete. He considers 

 the princi]3)es employed in this inquiry as presenting a solution of 

 many difficulties, and illustrating peculiarities appertaining to the 

 theory of logarithms of negative quantities ; and when applied to 

 geometry, as furnishing the means of tracing the form and developing 

 the properties of curves whose equations involve exponential quan- 

 tities. He also states, that by their means various differential and 

 other formulcfi, usually exhibited in treatises on logarithms, may be 

 rendered complete. An appendix is subjoined, containing several 

 examples of these applications of his principles. In the course of his 

 investigations, the author endeavours to explain the remarkable ano- 

 maly which frequently presents itself to the analyst of developments, 

 in which, upon substituting a particular value for the variable in each, 

 there is no approximation to numerical identity between the several 

 resulting series calculated to any number of terms, and the respective 

 functions which they ought to represent. He combats the paradoxical 

 opinion which has been advanced, that equations which in particular 

 instances were numerically false, were yet analytically true ; and ex- 

 plains the difficulty by reverting to the limitiitions inherent in the hy- 

 pothesis upon which the development is founded. He maintains, in 

 opposition to the opinions of Jean Bernouilli and D'Alembert, that 

 the logarithms of negative numbers are not in general the same as 

 their positives ; and hence infers, that negative numbers have occa- 

 sionally even real logarithms. The chief novelty of his system con- 

 sists in showing that any assigned quantity, relatively to a given 

 base, ha* an infinite number of orders of logarithms, and an infinite 

 number ot logarithms in each order. 



Another paper was read, intitled, " Experiments on the modulus 

 of torsion." By B. Bevan, Esq. ; communicated by the President. 



The object of the author in this paper is to ascertain the modulus 

 of torsion in different species of wood, and also of metals, deduced 

 from experiments on a large scale, which he conceives will furnish 

 many useful data applicable to practice by the mechanic and engineer. 



Care 



